Heat gas grill or coals. Mix Parmesan cheese and taco seasoning mix. Coat chicken with cheese mixture.
When ready to grill, carefully oil grill rack. Place chicken on gas grill over medium heat or on charcoal grill over medium coals. Cook 10 to 12 minutes, turning once or twice, until juice of chicken is clear when center of thickest part is cut (170°F). Cut each chicken breast in half crosswise; top each with half slice of cheese to melt. To heat dinner rolls, place rolls in foil on grill during last 6 to 7 minutes of cooking time until hot.
Spread mayonnaise on bottom halves of rolls. Top with lettuce leaf, cheese-topped chicken, salsa and top half of roll.